Group #49096272 — subgroups

Subgroups — #49096272

Order 35968.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22582871	C₂	1	1	yes	maximal, characteristic
2	#31823333	C₂	1	1	yes	maximal, characteristic
3	#31823226	C₂	1	4	yes	maximal
4	C₂×#97×C₂₈₁	C₂	1	1	yes	maximal, characteristic
5	#22582871	—	2	2	no	—
6	#22582871	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₂₈₁	—	2	2	no	—
8	C₂×C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
10	C₂×#17985117	—	2	2	no	—
11	C₂²×C₈×C₂₈₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
13	#97×C₂₈₁	—	2	2	no	—
14	C₂²×Dic₂₈₁	—	2	2	no	—
15	C₂²×Dic₂₈₁	—	2	2	no	—
16	C₄⋊Dic₂₈₁	—	2	2	no	—
17	C₄⋊Dic₂₈₁	—	2	4	no	—
18	C₄⋊Dic₂₈₁	—	4	4	no	—
19	C₄⋊Dic₂₈₁	C₂×C₄	1	4	yes	—
20	#17985117	—	2	8	no	—
21	C₂²×C₄×C₂₈₁	D₄	1	2	yes	—
22	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₈₁	—	2	2	no	—
24	C₂×C₈×C₂₈₁	—	2	2	no	—
25	C₂×C₈×C₂₈₁	Q₈	1	4	yes	—
26	C₄²×C₂₈₁	C₂×C₄	1	2	yes	—
27	C₄²×C₂₈₁	D₄	1	2	yes	—
28	C₂×Dic₂₈₁	—	2	4	no	—
29	C₂×Dic₂₈₁	—	4	2	no	—
30	C₂×Dic₂₈₁	—	4	2	no	—
31	C₂×Dic₂₈₁	—	4	2	no	—
32	C₂×Dic₂₈₁	—	4	4	no	—
33	C₂³×C₂₈₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₈₁	—	2	2	no	—
35	C₂×C₄×C₂₈₁	—	2	2	no	—
36	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₈₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₂₈₁	#41	1	2	yes	—
41	C₈×C₂₈₁	—	2	4	no	—
42	C₂²×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₂₈₁	#124	1	1	yes	characteristic
44	C₂²×C₂₈₁	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₂₈₁	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₂₈₁	#137	1	2	yes	—
47	C₄×C₂₈₁	—	2	4	no	—
48	C₄×C₂₈₁	C₄×Q₈	1	2	yes	—
49	C₄×C₂₈₁	C₂²⋊Q₈	1	2	yes	—
50	Dic₂₈₁	—	4	4	no	—
51	Dic₂₈₁	—	8	2	no	—
52	C₂×C₂₈₁	#391	1	1	yes	characteristic
53	C₂×C₂₈₁	#417	1	2	yes	—
54	C₂×C₂₈₁	#490	1	1	yes	characteristic
55	C₂×C₂₈₁	#491	1	1	yes	characteristic
56	C₂×C₂₈₁	#501	1	2	yes	—
57	C₂₈₁	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	281	1	no	maximal, Sylow
59	C₂×#125	—	281	1	no	—
60	C₂×#97	D₂₈₁	1	1	yes	characteristic
61	#384	—	281	1	no	—
62	#340	—	281	4	no	—
63	C₂²×C₈	C₂×D₂₈₁	1	2	yes	—
64	#125	—	281	4	no	—
65	#125	—	562	2	no	—
66	C₂×C₄⋊C₄	—	281	2	no	—
67	C₂×C₄⋊C₄	—	281	2	no	—
68	C₂×C₄⋊C₄	—	562	2	no	—
69	C₂×C₄⋊C₄	—	562	2	no	—
70	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
71	#97	—	2	2	no	—
72	C₂²×C₄	—	562	2	no	—
73	C₂²×C₄	—	562	2	no	—
74	C₂²×C₄	D₁₁₂₄	1	2	yes	—
75	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Dic₅₆₂	1	4	yes	—
79	C₄⋊C₄	—	281	4	no	—
80	C₄⋊C₄	—	562	8	no	—
81	C₄⋊C₄	—	562	2	no	—
82	C₄⋊C₄	—	562	4	no	—
83	C₄⋊C₄	—	1124	4	no	—
84	C₄²	C₄×D₂₈₁	1	2	yes	—
85	C₄²	C₂²⋊D₂₈₁	1	2	yes	—
86	C₂³	C₂×D₁₁₂₄	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	562	4	no	—
90	C₂×C₄	—	1124	2	no	—
91	C₂×C₄	—	1124	2	no	—
92	C₂×C₄	—	1124	2	no	—
93	C₂×C₄	—	1124	4	no	—
94	C₂×C₄	#17985115	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₅₆₂	1	2	yes	—
98	C₂×C₄	#17985138	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×#17985115	1	1	yes	characteristic
101	C₂²	#22582867	1	1	yes	characteristic
102	C₂²	#22582879	1	1	yes	characteristic
103	C₂²	#22582907	1	2	yes	—
104	C₂²	#22582912	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	1124	4	no	—
107	C₄	—	2248	2	no	—
108	C₄	C₄×Dic₅₆₂	1	2	yes	—
109	C₄	#22582858	1	2	yes	—
110	C₂	#31823369	1	1	yes	characteristic
111	C₂	#31823465	1	2	yes	—
112	C₂	#31823828	1	1	yes	characteristic
113	C₂	#31823832	1	1	yes	characteristic
114	C₂	#31823872	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582871

C₂

yes

maximal, characteristic

#31823333

C₂

yes

maximal, characteristic

#31823226

C₂

yes

maximal

C₂×#97×C₂₈₁

C₂

yes

maximal, characteristic

#22582871

—

#22582871

C₄

yes

complemented

C₂×C₄⋊Dic₂₈₁

—

C₂×C₄⋊Dic₂₈₁

C₂²

yes

—

C₂×C₄⋊Dic₂₈₁

C₂²

yes

—

C₂×#17985117

—

C₂²×C₈×C₂₈₁

C₂²

yes

—

C₂×C₄²×C₂₈₁

C₂²

yes

characteristic

#97×C₂₈₁

—

C₂²×Dic₂₈₁

—

C₂²×Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

C₂×C₄

yes

—

#17985117

—

C₂²×C₄×C₂₈₁

D₄

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

Q₈

yes

—

C₄²×C₂₈₁

C₂×C₄

yes

—

C₄²×C₂₈₁

D₄

yes

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂³×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

—

C₂×C₄×C₂₈₁

#41

yes

—

C₈×C₂₈₁

—

C₂²×C₂₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₁

#124

yes

characteristic

C₂²×C₂₈₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₈₁

AGL(1,ℤ/8)

yes

—

C₂²×C₂₈₁

#137

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₄×Q₈

yes

—

C₄×C₂₈₁

C₂²⋊Q₈

yes

—

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

#391

yes

characteristic

C₂×C₂₈₁

#417

yes

—

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

—

C₂₈₁

#1992

yes

characteristic, complemented, Sylow

#1992

—

281

maximal, Sylow

C₂×#125

—

281

—

C₂×#97

D₂₈₁

yes

characteristic

#384

—

281

—

#340

—

281

—

C₂²×C₈

C₂×D₂₈₁

yes

—

#125

—

281

—

#125

—

562

—

C₂×C₄⋊C₄

—

281

—

C₂×C₄⋊C₄

—

281

—

C₂×C₄⋊C₄

—

562

—

C₂×C₄⋊C₄

—

562

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

#97

—

C₂²×C₄

—

562

—

C₂²×C₄

—

562

—

C₂²×C₄

D₁₁₂₄

yes

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₅₆₂

yes

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

1124

—

C₄²

C₄×D₂₈₁

yes

—

C₄²

C₂²⋊D₂₈₁

yes

—

C₂³

C₂×D₁₁₂₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

562

—

C₂×C₄

—

1124

—

C₂×C₄

—

1124

—

C₂×C₄

—

1124

—

C₂×C₄

—

1124

—

C₂×C₄

#17985115

yes

—

C₂×C₄

C₂×C₄×D₂₈₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

C₂×C₄

C₂×Dic₅₆₂

yes

—

C₂×C₄

#17985138

yes

—

C₈

—

100

C₂²

C₂×#17985115

yes

characteristic

101

C₂²

#22582867

yes

characteristic

102

C₂²

#22582879

yes

characteristic

103

C₂²

#22582907

yes

—

104

C₂²

#22582912

yes

—

105

C₄

—

106

C₄

—

1124

—

107

C₄

—

2248

—

108

C₄

C₄×Dic₅₆₂

yes

—

109

C₄

#22582858

yes

—

110

C₂

#31823369

yes

characteristic

111

C₂

#31823465

yes

—

112

C₂

#31823828

yes

characteristic

113

C₂

#31823832

yes

characteristic

114

C₂

#31823872

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₂₄
commutator	C₂×C₄×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂×#97×C₂₈₁	C₂
socle, prime radical	C₂³×C₂₈₁	C₂×D₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×#97	D₂₈₁
hypocenter	C₂₈₁	#1992
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₁₂₄

commutator

C₂×C₄×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂×#97×C₂₈₁

C₂

socle, prime radical

C₂³×C₂₈₁

C₂×D₄

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₂×#97

D₂₈₁

hypocenter

C₂₈₁

#1992

LCS

G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
6. #22582871	C₄	semidirect
57. C₂₈₁	#1992	semidirect

H ≅ G/N

type

6. #22582871

C₄

semidirect

57. C₂₈₁

#1992

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#22582871	2
#31823333	2
#31823226	2
C₂×#97×C₂₈₁	2
#1992	281

subgroup

index

C₂×#97×C₂₈₁

281

Subgroups — #49096272

Conjugacy classes of subgroups

Complements

Maximal subgroups